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Nearby Supernova Factory : Calibration of SNIFS data
and spectrophotometric Type Ia supernovae light curves.
Rui da Silva Pereira
To cite this version:
Sp´ecialit´e
Astrophysique et M´ethodes Associ´ees
Pr´esent´ee par
M. Rui DA SILVA PEREIRA en vue de l’obtention du grade de
DOCTEUR de L’UNIVERSIT ´E PARIS.DIDEROT (Paris 7)
Nearby Supernova Factory :
´
Etalonnage des donn´ees de SNIFS et courbes de lumi`ere
spectrophotom´etriques de supernovæ de type Ia.
Calibration of SNIFS data and spectrophotometric
Type Ia supernovæ light curves.
Soutenue le 5 d´ecembre 2008 devant le jury compos´e de :
Pierre ANTILOGUS Directeur de th`ese
James BARTLETT Pr´esident du jury
Ariel GOOBAR Rapporteur
Ana MOUR ˜AO Examinateur
M´ario PIMENTA Examinateur
Se v˜ao da lei da Morte libertando, Cantando espalharei por toda parte, Se a tanto me ajudar o engenho e arte.
In no particular order, I would like to thank:
• First of all, Pierre Antilogus, for his flawless supervision. It was a pleasure to work with him during the last 3 years, and I’ll miss his bursts of laughter echoing around the lab :) • Jim Bartlett, for accepting to be the president of my thesis defense committee, Ariel
Goobar and Jim Rich for having the patience and the availability to referee my thesis in such a short time frame, and Ana Mour˜ao and M´ario Pimenta for also accepting to be members of the defense committee;
• All the members of the SNfactory collaboration, the folks of the Lyon and Berkeley teams, and especially the ones which whom I worked most closely in Paris, Gabriele Garavini, S´ebastien Gilles, Stephen Bailey and Chao Wu;
• The Supernovæ group at the LPNHE, particularly Nicolas Regnault, Pierre Astier and Julien Guy, for their patience and availability to answer my questions about poloka or SALT;
• The people of the LPNHE, Ursula Bassler and Jos´e Ocariz, two perfect thesis “god-mother/father”, and all the “jeunes” who came and went during these years and shared meals, doubts, laughs or brain-dead emails: Delphine Guide, Bruno Marcos, Guillaume Th´erin, Emmanuel Busato, Emannuel Hornero, Claire Juramy, Sylvain Baumont, Diego Terront, Luz Guevara, Alejandro, Henso, Homero, Andr´es and all the other members of the Venezuelan troupe, Florent Fayette, J´erome Glisse, Andrzej Si´odmok, Taia Kronborg, Nicolas Fourmanoit, Marc Dhellot and of course, my two favorite italians, Pietro and Stefania;
• The guys of the Meudon M2 2004 promo, Arnaud, Benoit(s), Marc, Fabio, Ferras and Xavier(s), for all the shared beers and good moments; and Jacqueline Plancy: I would still be wrestling with university bureaucracy if it wasn’t for her;
• All of those who I (momentarily) forgot to include in this list.
I would also like to acknowledge Ph.D. financial support from Funda¸c˜ao para a Ciˆencia e Tecnologia, grant number SFRH/BD/18494/2004.
La derni`ere d´ecade a ´et´e une ´epoque marqu´ee par de nombreux changements en cosmologie. De nouvelles m´ethodes observationnelles, avec une pr´ecision grandissante, ont chang´e notre vision de l’univers et la fa¸con dont nous l’´etudions. Un mod`ele en accord avec les observations est donc apparu (§1): un mod`ele o`u l’univers subit une expansion acc´el´er´ee, emport´ee par une ´
energie sombre d’origine inconnue, et o`u la majeure partie de la mati`ere est sous la forme d’une mati`ere sombre invisible. Actuellement, environ 95% du contenu total en ´energie de l’univers est d’origine incertaine.
Les supernovæ de type Ia (SNe Ia) (§ 2.3) repr´esentent un des outils observationnels `a notre disposition. G´en´eralement accept´ees comme ´etant l’explosion thermonucl´eaire d’une naine blanche, ayant atteint sa masse critique apr`es accr´etion de masse `a partir d’un compagnon dans un syst`eme binaire, les SNe Ia sont des ´ev`enements extrˆemement lumineux visibles `a des distances cosmologiques. Les SNe Ia ont aussi une luminosit´e remarquablement homog`ene; c’est pourquoi des m´ethodes d’´etalonnage ont ´et´e d´evelopp´ees, pour permettre leur utilisation comme outils pour la mesure de distances (des chandelles standards), permettant de sonder la g´eom´etrie et la dynamique de l’univers et de ses composants. Les SNe Ia sont un des piliers de la cosmologie observationnelle moderne, avec les mesures du fond diffus cosmologique et de la formation de grandes structures dans l’univers.
L’analyse cosmologique `a partir des SNe Ia doit s’appuyer sur l’observation d’objets mesur´es `
a grands et petits d´ecalages vers le rouge (z). La d´etermination des param`etres cosmologiques est bas´ee sur une comparaison entre des ´echantillons de ces deux types d’objets, cependant tandis que le nombre d’objets mesur´es `a grands d´ecalages vers le rouge a augment´e progressivement dans les derni`eres ann´ees, la quantit´e d’objets proches mesur´es de fa¸con consistante est toujours faible. Ceci constitue actuellement l’une des plus grandes sources d’erreurs syst´ematiques dans les analyses de cosmologie bas´ees sur les SNe Ia.
Le cadre de travail
Cette th`ese s’inscrit dans le cadre de l’exp´erience SNfactory (Partie II), une collaboration franco-am´ericaine, qui a pour objectif l’observation d’environ 200 SNe Ia proches (0.03 < z < 0.08) issues d’une recherche propre et suivies grˆace `a un spectrographe de champ int´egral sp´ecialement con¸cu pour l’´etude de SNe Ia, snifs (§ 4). Les s´eries temporales de spectres de SNe Ia, ´etalonn´ees de mani`ere absolue en flux, sont des donn´ees uniques appropri´ees `a une analyse soit spectroscopique, soit spectrophotom´etrique. Cet ensemble de donn´ees est ´egalement fondamental pour un usage plus pr´ecis des SNe Ia dans un cadre cosmologique, et aussi pour une meilleure compr´ehension de la physique des explosions de supernovæ.
ACKNOWLEDGEMENTS
R´
esum´
e du travail et r´
esultats
La voie photom´etrique de snifs (§ 4.2) est constitu´ee des cam´eras (CCD) qui permettent l’observation du champ autour de la supernova, en parall`ele avec l’acquisition des spectres. Ces cam´eras permettent d’acqu´erir des donn´ees essentielles pour la calibration absolue en flux des spectres, dans la mesure o`u elles permettent la d´etermination de l’att´enuation atmosph´erique diff´erentielle entre nuits non-photom´etriques.
La premi`ere partie de ce travail a ´et´e consacr´ee `a l’´etalonnage des donn´ees brutes issues de la voie photom´etrique (§ 8), de la mise en place de proc´edures de “nettoyage” de lumi`ere parasite jusqu’`a la caract´erisation des divers param`etres de la cam´era photom´etrique. L’objectif de ces ´etudes est d’obtenir des rapports photom´etriques (§9) entre nuits. Cette premi`ere partie a abouti `a la cr´eation d’une chaˆıne de traitement et d’extraction des rapports photom´etriques, qui a ´et´e int´egr´ee dans le traitement centrale de SNfactory. La qualit´e des rapports photom´etriques produits a ´et´e ´evalu´ee, et ses erreurs syst´ematiques ont ´et´e estim´ees comme ´etant inf´erieures `a 2%.
Par la suite, les images d’acquisition issues du pointage du t´elescope (un sous-produit de la voie photom´etrique) ont ´et´e utilis´ees pour la cr´eation de courbes de lumi`ere d’une seule couleur (§ 10). Ces courbes de lumi`ere ont ´et´e ajust´ees avec SALT, le programme d’ajustement de courbes de lumi`ere cr´ee par la collaboration SNLS, qui est utilis´e pour la standardisation des observations de SNe Ia `a grands et petits d´ecalages vers le rouge. Des courbes cr´ees, 73 on ´et´e utilis´ees dans une premi`ere ´etude des distributions de phase (la date de d´ecouverte de la super-nova par rapport au maximum de sa courbe de lumi`ere) et de “stretch” (le facteur d’´etirement utilis´e par SALT pour d´ecrire la relation plus brillant – plus lent trouv´e sur les courbes de lumi`ere des SNe Ia). Les deux distributions sont en accord avec des donn´ees pr´ec´edemment publi´ees, ce qui d´emontre que cet ´echantillon d’objets est appropri´e pour une analyse cosmologique. Un biais dans les r´esultats de SALT a cependant ´et´e observ´e, pour les courbes de lumi`ere ne poss`edent pas des mesures avant le maximum de luminosit´e.
Avec les rapports photom´etriques calcul´es pr´ec´edemment, la chaˆıne d’´etalonnage en flux a ´
et´e mise en oeuvre sur les spectres de six ´etoiles standards spectrophotom´etriques (§11.2). Les courbes de lumi`ere en r´esultant ont permis de d´eterminer la pr´ecision actuelle de l’´etalonnage en flux de SNfactory dans les nuits non-photom´etriques (∼ 5%) et aussi de confirmer la consistance en couleur des spectres extraits. Les courbes de lumi`ere de 30 supernovæ on ´et´e par la suite synth´etis´ees et ajust´ees avec SALT. Ces courbes de lumi`ere repr´esentent ainsi les premi`eres courbes de lumi`ere jamais issues de donn´ees spectrophotom´etriques (§11.3).
cette dispersion due `a la K-correction dans les analyses cosmologiques avec SNe Ia. Un ´etude exploratoire pour l’identification et la compr´ehension d’objets particuliers, par l’utilisation de courbes de lumi`ere de filtres ´etroits plac´es sur des particularit´es spectrales, a aussi ´et´e effectu´ee (§12.3). Nous pouvons conclure que la qualit´e actuelle de l’extraction des spectres et le nombre d’objets disponibles ne permettent pas la construction d’une m´ethode robuste de rejet de points aberrants, n´ecessaire pour une ´etude de ce genre. N´eanmoins une ´etude qualitative de certaines courbes de lumi`ere ´etroites a sugg´er´e l’existence possible de sous-familles inclues dans la famille particuli`ere des SNe Ia 91T-like (surlumineuses).
Introduction 1
I Observing the Universe 5
1 Big Bang cosmology 7
1.1 Homogeneity and isotropy . . . 7
1.2 FLRW models in general relativity . . . 8
1.2.1 Robertson-Walker (RW) metric . . . 8
1.2.2 Einstein equations . . . 8
1.2.3 Perfect fluid approximation . . . 9
1.2.4 Friedmann-Lemaˆıtre (FL) equations of motion . . . 9
1.3 The expanding universe . . . 10
1.3.1 Cosmological redshift . . . 10
1.3.2 Hubble’s discovery . . . 11
1.3.3 Universe composition evolution . . . 11
1.3.4 Cosmological parameters . . . 13
1.3.5 Cosmological observables . . . 16
1.4 The ΛCDM concordance model . . . 18
1.4.1 Extension models . . . 19
2 Observational cosmology 21 2.1 Cosmic Microwave Background . . . 21
2.2 Large Scale Structure . . . 22
2.3 Supernovæ . . . 24
2.3.1 Classification and origins . . . 24
2.3.2 SNe Ia homogeneity - standard candles . . . 27
2.3.3 SNe Ia heterogeneity - peculiar objects . . . 30
2.3.4 SNe Ia as a dark energy probe . . . 32
2.3.5 The need for nearby SNe Ia . . . 37
II The SNfactory project 41 3 A new approach to SNe observation 43 3.1 Improving our knowledge of SNe Ia . . . 43
CONTENTS
4 The SuperNova Integral Field Spectrograph 47
4.1 Spectroscopic channels . . . 48
4.2 Photometric channel . . . 51
4.2.1 Multi-filter . . . 52
4.3 Data taking . . . 53
4.4 Problems with SNIFS . . . 54
5 The observational implementation 57 5.1 Search . . . 57
5.2 Screening . . . 58
5.3 Spectroscopic followup . . . 59
5.4 Interesting SNfactory objects . . . 61
5.4.1 SN2005gj . . . 61
5.4.2 SN2006D . . . 61
5.4.3 SNF20070803-005 . . . 63
5.4.4 SNF20070825-001 . . . 63
5.4.5 SNF20080720-001 . . . 65
III From photons to bits - SNfactory’s data acquisition & calibration 67 6 Data acquisition rationale 69 6.1 A night’s run schedule . . . 69
6.2 Science exposures . . . 69 6.2.1 Spectrum datacube . . . 70 6.2.2 Acquisition exposure . . . 70 6.2.3 Multi-filter exposure . . . 70 6.3 Calibration exposures . . . 72 6.3.1 Bias . . . 72 6.3.2 Dark . . . 72 6.3.3 Flat field . . . 72 6.3.4 Continuum . . . 72 6.3.5 Arc . . . 72
7 Spectra flux calibration 73 7.1 Atmospheric extinction and photometricity . . . 73
7.1.1 Spectroscopic channel . . . 76
7.1.2 Photometric channel . . . 76
7.2 Photometric night . . . 76
7.3 Non-photometric night . . . 77
7.3.1 Photometric channel ratios . . . 78
7.3.2 The gray hypothesis . . . 79
8 Photometric channel calibration 81 8.1 Pickup noise . . . 81
8.2 Flat fielding . . . 83
8.3 Extra light and light leaks . . . 85
8.3.1 Hawaii intervention . . . 87
8.3.3 Fringing . . . 89
8.4 Shutter latency . . . 92
8.5 Gain variation . . . 93
8.5.1 Gain determination . . . 94
8.5.2 Absolute variation and correlation with flux . . . 95
8.5.3 Relative variation . . . 99
8.5.4 Guiding CCD gain . . . 99
8.6 Spectro/photometric channels intercalibration . . . 101
8.7 Filter zero points . . . 101
8.8 MLA object position determination . . . 103
9 Photometric ratios extraction 107 9.1 Pipeline implementation . . . 107 9.2 Pipeline flow . . . 107 9.2.1 ACQ vs. MF . . . 108 9.2.2 Filter cutting . . . 108 9.2.3 DbImage . . . 109 9.2.4 Object identification . . . 109 9.2.5 Aperture photometry . . . 110 9.2.6 Image alignment . . . 111
9.2.7 Stacking and reference catalog . . . 111
9.2.8 Image subtraction - photometric ratios . . . 114
9.2.9 Photometric ratios parsing . . . 115
9.2.10 PSF photometry . . . 115
9.2.11 Acquisition light curves . . . 116
9.3 Quality benchmarking . . . 116
9.3.1 Optimal aperture size for photometry . . . 117
9.3.2 Seeing determination . . . 118
9.3.3 Error estimation . . . 120
9.3.4 Ratios statistics and comparison . . . 125
IV From bits to light curves - SNfactory’s data analysis 129 10 Acquisition exposures’ analysis 131 10.1 Bessell V light curves . . . 131
10.1.1 SALT fits . . . 131
10.2 SNfactory sub-dataset phase and stretch distribution . . . 134
10.2.1 SCP Union compilation comparison . . . 135
10.3 SALT bias . . . 136
11 Spectrophotometric light curves 141 11.1 Gray extinction hypothesis testing . . . 141
11.2 Fundamental standard calibrators . . . 142
11.3 Supernovæ sub-sample . . . 145
11.3.1 SALT2 fits . . . 146
CONTENTS
12 Hubble diagram 155
12.1 Spectrophotometric nearby Hubble diagram . . . 155
12.1.1 “Peculiar” objects . . . 157
12.2 K-correction free nearby Hubble diagram . . . 160
12.3 Sharp filter light curves . . . 166
12.3.1 Pull analysis . . . 168
12.3.2 91T-like qualitative light curve analysis . . . 169
Conclusion & perspectives 173 V Appendix 175 A SnfPhot pipeline 177 A.1 Software call . . . 177
A.2 GUI . . . 178
A.3 Documentation . . . 178
B Extra tables 179 B.1 Spectrophotometric vs. Bessell V light curves . . . 179
B.2 K-correction free SALT2 fits . . . 181
B.3 Hubble diagram results without LC points’ error scaling . . . 183
C SNe Ia spectrophotometric light curves 185 C.1 [BVR]SNf SALT2 fits . . . 185
C.2 Sharp Ca + Si2 SALT2 fits . . . 191
C.3 Sharp Si + Si2 + Si3 SALT2 fits . . . 197
The last decade has been an eventful one for cosmology. New observational methods, with ever increasing precision, changed our vision of the universe and the way we study it.
An observationally consistent model has emerged. One of an universe in accelerated expan-sion, driven by a dark energy due to an elusive cosmological constant, and where most of the matter is in the form of an invisible dark matter. Such is the scenario cosmology faces today: roughly 95% of the total energy distribution of the universe is of uncertain origin. These are interesting times indeed!
One of the observational tools at our disposal, is the Type Ia supernovæ (SNe Ia). Commonly accepted as the thermonuclear explosion of a white dwarf star, who reached its critical mass due to matter accretion from a binary companion, these events are extremely bright, making them visible up to cosmological distances. They are also remarkably homogeneous: standardization methods have been developed that effectively allow their usage as a distance ruler (a standard candle) to probe the geometry and dynamics of the universe, and hence its constituents. They are one of the cornerstones of modern observational cosmology, along with measurements of the cosmic microwave background radiation and large scale structure.
The last years have seen the establishment of several supernovæ surveys, which aim to collect the biggest possible dataset of SNe Ia observations, from nearby and distant events (at low or high redshift). The determination of the cosmological parameters rests on a comparison between these two samples, hence the interest of having high quality, well sampled data on both distance ranges. That is not however the case. Nearby supernovæ (at distances where their observed luminosity is independent of the cosmological model), are more difficult to find, due to the larger areas of sky that need to be scanned. Therefore, while the statistics of high redshift SNe Ia increase, the number of well observed, consistent datasets of nearby supernovæ is still small. They represent today one of the main sources of systematic errors on the cosmological analysis using supernovæ.
Among the current low redshift surveys built to tackle this problem, is the SNfactory project. It aims to observe around 200 nearby SNe Ia with high quality spectral data, originated from a dedicated integral field unit spectrograph, snifs. The resulting dataset is unique in the sense that it consists of spectral time series of SNe Ia, suitable either for spectroscopic or spectrophotometric analysis. It is at the same time a fundamental dataset for an higher precision usage of SNe Ia as dark energy probes, and for a better understanding of the physics behind supernovæ explosions. I joined the SNfactory collaboration on September 2005, and this manuscript aims to group the ensemble of the work and the contributions to the project made during my thesis. It was mainly focused on the development and implementation of a calibration and extraction pipeline for the snifs’ photometric channel data, essential for the flux calibration of the spectra. I also took part on the ongoing scientific analysis effort, for which I contributed with a novel way of using snifs’ unique dataset on a cosmological fit point of view.
INTRODUCTION
Part I consists in an introduction to cosmology. In Chapter 1 the fundamental equations of cosmology will be presented, as well as the cosmological parameters and observables, leading up to the current standard cosmological model. Chapter 2 will then introduce the three main axis of observational cosmology: the cosmic microwave background and large scale structure measurements are briefly described, while bigger emphasis is given to the supernovæ. Their classification, origins and photometric and spectral homogeneity vs. peculiarity are discussed, as well as their current usage as dark energy probes.
PartIIpresents the SNfactory project and its unique observational approach. In Chapter 3, the scientific goals of SNfactory and the spectrophotometry concept are introduced. Chapter4 describes the snifs instrument, the integral field spectrograph built by the collaboration for dedicated SNe Ia observations, its division into spectroscopic and photometric channels, and the remote data taking procedure. In Chapter 5 the different components of the project are detailed: the supernovæ search survey using the QUEST-II camera, and subsequent screening and spectroscopic followup made using snifs. Statistics from the current SNfactory sample are shown as well as a number of interesting (peculiar) objects observed by SNfactory.
PartIIIdetails the data acquisition and calibration procedures for snifs, with special focus on the photometric channel data calibration, extraction and its usage on the spectra flux cali-bration scheme. In Chapter 6 a typical (remote) observation night is explained, as well as the different kinds of data acquired by snifs during each run. Chapter 7 describes the full snifs spectral calibration procedure, starting from the raw spectral data up until the flux calibrated spectra, and on the role of the photometric channel on the flux calibration of non-photometric nights, using between nights’ photometric ratios. In Chapter 8 all the different aspects of the calibration of the raw photometric channel exposures are detailed. Chapter 9 explains the full pipeline work-flow for the extraction of the photometric ratios from the photometric channel data, and presents quality benchmarks.
Big Bang cosmology
There is a theory which states that if ever anyone discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable.
There is another which states that this has already happened.
Douglas Adams
Cosmology is the science of the universum, the “combination of all things into a whole”. It aims to explain the origin and evolution of the entire contents of the universe and the underlying physical processes.
Throughout history, mankind saw the accumulation of astronomical evidences contradicting further and further its anthropocentric beliefs. They took us from being in the center of the “creation” up to inhabiting an average-sized planet orbiting an average-sized star in the periph-ery of a average-sized rotating galaxy. A galaxy flying towards an unknown goal in the middle of billions of similar galaxies, in an universe with a center which is everywhere and nowhere at the same time, where no location is more special than any other.
Such accumulated evidences formed thus our current understanding of the universe: an evolving entity whose birth, in a singular moment of infinite density, pressure and spacetime curvature, and growth is explained by the hot Big Bang theory. This theory lies itself on the theoretical framework of Einstein’s general relativity and on well established observational evidences.
1.1
Homogeneity and isotropy
Inherent in the foundations of this cosmological theory is the cosmological principle, first stated by Copernicus. It hypothesizes that the universe is spatially homogeneous and isotropic, that is, spatially invariant under translation and rotation, thus making all positions fundamen-tally equivalent.
Specifically, by homogeneous we mean that (at large scales, bigger than 100 Mpc1) the universe looks the same at every point and by isotropic that it looks the same in all directions. These assumptions are nowadays well confirmed respectively by the large-scale distribution of galaxies (§ 2.2) and the near-uniformity of the CMB temperature (§ 2.1).
1
CHAPTER 1. BIG BANG COSMOLOGY
Imposing these symmetries2 to the universe at large scale, we can thus study its evolution.
1.2
FLRW models in general relativity
1.2.1 Robertson-Walker (RW) metric
Under the assumption of the cosmological principle, our spacetime metric can be written in terms of an invariant four dimensions geodesic metric gµν which follows
ds2 = gµνdxµdxν , (1.1)
that is, relates length measurements ds to a chosen coordinate system dx (and thus is an intrinsic description of space), as the so-called Robertson-Walker metric
ds2 = dt2− a2(t) dr2 (1 − kr2) + r 2 dθ2+ sin2θdφ2 . (1.2)
We are assuming the speed of light is equal to unity (c = 1) and: t is the proper time, r, θ, φ are comoving (spherical) spatial coordinates, a(t) is the scale factor (normalized by a(t0) ≡ 1)3,
and the constant k characterizes the spatial curvature of the universe. k = {−1, 0, 1} correspond respectively to open (saddle-like), spatially flat and closed (3-sphere) universes.
1.2.2 Einstein equations
The cosmological equations of motion are derived from the Einstein (1917)field equations
Rµν−
1
2gµνR = 8πGTµν− Λgµν , (1.3)
where G is Newton’s gravitational constant, and which relate the curvature, represented by the Ricci tensor Rµν to the energy-momentum tensor Tµν4. The Λgµν term is the cosmological
constant (Λ) term and was added by Einstein since he was interested in finding static solutions, both due to his personal beliefs and in accordance to the astronomical data available at the time, but which his initial formulation did not allowed. This effort however was unsuccessful: the static universe described by this theory was unstable, and both de Sitter’s solution of (1.3) for empty universes with Λ > 0, and measurements of the radial velocities of distant galaxies using the redshift (§1.3.1) of their spectra bySlipher (1924), showed that our universe is in fact not static, which led Einstein to abandon the cosmological constant.
Despite Einstein’s misguided motivation for introducing the cosmological constant term, there is nothing wrong (i.e. inconsistent) with the presence of such a term in the equations. We will see in §1.2.4what this term may represent in modern cosmology.
2
The homogeneity and isotropy are symmetries of space and not of spacetime. Homogeneous, isotropic space-times have a family of preferred three-dimensional spatial slices on which the three-dimensional geometry is homogeneous and isotropic.
3
Throughout this document, this normalization will be implicit and the0 subscript will always represent the present epoch.
4
1.2.3 Perfect fluid approximation
The cosmological principle gives rise to the picture of the universe as a physical system of “cosmic fluid” whose fundamental particles are galaxies. We thus assume that the matter content of the universe is a perfect fluid with energy-momentum tensor
Tµν = (p + ρ) uµuν− pgµν , (1.4)
where the metric gµν is described by (1.2), ρ is the energy density, p the isotropic pressure and
u = (1, 0, 0, 0) the velocity 4-vector for the isotropic fluid in comoving coordinates.
1.2.4 Friedmann-Lemaˆıtre (FL) equations of motion
By substitution of the perfect fluid energy-momentum tensor (1.4) and the RW metric (1.2) on Einstein’s equations (1.3), we can obtain the so-called Friedmann-Lemaˆıtre equations
H2≡ ˙a a 2 = 8πGρ 3 − k a2 + Λ 3 (1.5) ¨ a a = − 4πG 3 (ρ + 3p) + Λ 3 , (1.6)
which relate the dynamics of the universe (the time variation of the scale factor, ˙a) with the pressure and energy density of its constituents. H(t) is the Hubble parameter. Nowadays the Λ term is commonly thought as a measure of the energy density of the vacuum5, or more generally of a dark energy, so it can be represented with the other pressure and density terms6, as
Λ = 8πGρΛ= −8πGpΛ . (1.7)
Eq. (1.5) has a classical mechanical analog if we assume Λ = 0. Looking at −ak2 as a “total
energy”, then we see that the evolution of the universe is governed by the relation between the potential energy (8πGρ3 ) and the kinetic (a˙a) terms, making it expand or contract up to a fate determined by the curvature constant k. For k = 1 the universe recollapses in a finite time, whereas for k = {0, −1} it expands indefinitely. These conclusions can be altered when Λ 6= 0 (or more generally with some component with (ρ + 3p) < 0).
From equations (1.5) and (1.6) one can obtain the conservation equation (in agreement with the first law of thermodynamics), for the components of the universe
˙
ρ = −3H(ρ + p)
= −3Hρ(1 + w) , (1.8)
which expresses the evolution of energy density with time (its a-dependence). wi ≡ ρpii is the
equation of state parameter for any i component of the universe (cf. §1.3.3).
Armed with this theoretical background we are now able to apply our knowledge to study the evolution of the universe.
5
However this, as some (Kolb 2007) authors prefer to call it, cosmoillogical constant, is still one of the unresolved puzzles in fundamental physics, in view of the incredible fine-tuning problem which represents the several orders of magnitude difference between its present value and the value predicted by quantum field theory (Zel’Dovich
1967;Carroll 2001;Bousso 2008).
6
CHAPTER 1. BIG BANG COSMOLOGY
1.3
The expanding universe
1.3.1 Cosmological redshift
Most methods of measuring the composition of the universe rely on various observations of electromagnetic radiation. It is therefore important to discuss how the properties of light detected from a source are connected to a certain cosmological model.
By observing the spectra of starlight of nearby galaxies we can identify known emission lines and compare their wavelengths λ to the (known) emitted ones. These may appear displaced towards the blue or the red part of the spectrum, respectively blushifted or redshifted, in what may be interpreted as a Doppler effect in flat spacetime. That means that these galaxies are moving towards or away from us, with a velocity v related to the shift in wavelength ∆λ by the Doppler formula
z ≡ ∆λ
λ '
v
c , (1.9)
where z is the redshift and the second equality is valid for objects in our “neighborhood” (v c). If we have now a particle moving in a spacetime geometry described by the time dependent RW metric (1.2), its energy will change, similarly as it would if it moved in a time-dependent potential. For a photon whose energy is proportional to frequency, that change in energy is the cosmological redshift. We consider a photon emitted from a galaxy at comoving coordinate r = R at time t. It will travel in a geodesic ds2 = 0, so that in the time between emission and reception the photon traveled a spatial coordinate distance (in the spatially flat case)
R = Z t0
t
dt
a(t) . (1.10)
If we suppose instead that a series of pulses are emitted with frequency ω = 2πδt, the time interval between the pulses at reception δt0 can be calculated from (1.10) since all pulses travel
the same spatial coordinate separation R: Z t0+δt0 t+δt dt a(t) = R = Z t0 t dt a(t) , (1.11) and consequently δt0 a(t0) − δt a(t) = 0 , (1.12) which means ω0 ω = λ λ0 = a(t) a(t0) . (1.13)
Using (1.9) and a(t0) ≡ 1 we can write the definition of cosmological redshift
1
1 + z ≡ a(t) , (1.14)
1.3.2 Hubble’s discovery
By 1929, Hubble had obtained the distance to 18 nearby spiral galaxies (his “nebulæ”) and combined his results with the velocities measured by Slipher (1924). He found out that they were receding away from us with velocity given by v = cz, which increased linearly with distance (Fig. 1.1)
v = H0r . (1.15)
This is the Hubble’s law, with H0 the Hubble constant (the present value of the Hubble
parameter, introduced in § 1.2.4). Its message is that the universe is expanding and a static universe, where the galaxies should move about randomly is ruled out. The expanding universe, in this comoving coordinates’ frame where all galaxies rush away from each other, is interpreted not by a change in position coordinates but by an ever-increasing metric – the expansion of space itself.
Figure 1.1: Hubble’s original diagram relating the recession velocity of the “nebulæ” with their distance from us. The two lines use different corrections for the Sun’s motion. He found h ∼ 0.5. FromHubble (1929).
Hubble’s constant can be written as
H0 ≡ 100 h km s−1Mpc−1 , (1.16)
with recent measurements (Freedman et al. 2001) finding h = 0.72 ± 0.08 .
1.3.3 Universe composition evolution
CHAPTER 1. BIG BANG COSMOLOGY
From (1.8) and for a wi constant, we can write the time evolution of the energy density
˙ ρi ρi = −3˙a a(1 + wi) (1.17) ρi = ρi0 a a0 −3(1+wi) (1.18) ρi ∝ a−3(1+wi) (1.19) ∝ (1 + z)3(1+wi) , (1.20)
and using (1.5), (1.19) and the fact that, since at early times a is small and the curvature term can be neglected, we write the time evolution of the scale factor for a single component universe as
a(t) ∝ t
2
3(1+wi) . (1.21)
So, for the three most common barotropic (with pressure linearly proportional to density) fluids present in the universe, we can divide the history of the universe (Fig. 1.2) into three epochs: -1 0 1 2 3 4
Log [1+z]
-48 -44 -40 -36Log [energy density (GeV
4
)]
radiation
matter
dark energy
Figure 1.2: Evolution of radiation, matter, and dark energy densities with redshift. FromFrieman et al. (2008a).
Radiation-dominated In the early hot and dense universe, it is appropriate to assume an equation of state corresponding to a gas of radiation (or relativistic particles) for which wr= 13, leading to
ρr ∝ a−4 ; a(t) ∝ t
1
2 . (1.22)
We can see that the radiation density decreases with the forth power of the scale factor. Three of these powers are accounted for by the volume increase during the expansion, while the forth one comes from the cosmological redshift.
Matter-dominated At relatively late times, non-relativistic matter eventually dominates the energy density over radiation. It’s a pressureless gas of particles with temperatures much smaller than their mass for which wm = 0, and
ρm∝ a−3 ; a(t) ∝ t
2
Vacuum energy-dominated If there is a dominant source of vacuum energy that acts as a cosmological constant according to (1.7), its equation of state is wx = wΛ= −1, so
ρΛ∝ constant ; a(t) ∝ eHt , (1.24)
and the universe expands exponentially. However the vacuum equation of state does not need to be the w = −1 of Λ, neither it needs to be constant. That’s the more general possibility of a dynamical vacuum energy (§ 1.4.1) of the type
wx(a) = w0+ (1 − a)wa . (1.25)
Recent measurements (Komatsu et al. 2008), constraint respectively the dark energy constant and time-dependent equations of state (for a flat universe) by
−0.11 <1 + w < 0.14 (95% CL) −0.38 <1 + w0 < 0.14 (95% CL)
(For the sake of completeness, assuming (against all the major evidences (§2.1)) that our universe has k 6= 0 and we have a null dark energy parameter, we would have instead a curvature-dominated epoch, with
a(t) ∝ t . (1.26)
)
1.3.4 Cosmological parameters
We can now define a set of parameters which will describe the contents and their changes throughout the expansion history of the universe.
Energy densities
From the Friedmann equation (1.5), for any value of the Hubble parameter H there is a critical value of the density that would be required in order to make the spatial geometry of the universe flat, k = 0:
ρc ≡
3H2
8πG , (1.27)
which corresponds on the present epoch to
ρc(t0) = 1.88 h2× 10−26 kg m−3
= 2.78 h−1× 1011 M / h−1Mpc 3
= 11.26 h2 protons m−3 ,
or ∼ 6 protons per cubic meter, a rather dilute fluid. Since that the observed typical mass for a galaxy is in the order of 1011 ∼ 1012 M
(solar masses), and their separations in the order
of a Mpc, we can see that the universe cannot be far away from the critical density. Beyond this critical density, the universe’s geometry becomes closed, and the amount of matter provides enough gravitational attraction to stop and reverse the expansion.
The critical density can be used to set a natural scale for the density of the universe, by defining the density parameter
Ω ≡ ρ ρc
CHAPTER 1. BIG BANG COSMOLOGY
We can thus rewrite the Friedmann equation as Ω − 1 = k
a2H2 , (1.29)
with (for the i components, radiation, matter and dark energy)
Ω =X i Ωi= X i 8πGρi 3H2 . (1.30)
It is interesting to note the direct relation given by (1.29) between energy content and curvature of the universe: Ω {> 1, < 1, = 1} gives us respectively closed, open or spatially flat universes. Specifically stating the different contributions to the energy density today, we can write the “cosmic sum” which determines the overall sign of the curvature7
Ω0= Ωr+ Ωm+ Ωx=
k
H02 + 1 . (1.31)
Deceleration parameter
Looking now at the second Friedmann equation (1.6), we can define a new parameter, the so-called deceleration8 parameter
q0 ≡ − ¨ a a0H02 = 1 2 X i Ω0i(1 + 3wi) (1.32) = Ωr+ Ωm 2 + Ωx 2 (1 + 3wx) , (1.33)
which relates the dynamics of the universe with its contents. q0 = 0 would represent a static
universe, whether a positive or negative value means the expansion is decelerating or accelerating. For a flat universe, wx has to be < −13 for the latter case to happen.
Hubble parameter
The (time dependent) Hubble parameter already introduced in § 1.2.4can be related to its present day value (Hubble’s constant). Substituting (1.28) on (1.18) we obtain
ρi = ρcΩ0a−3(1+wi) , (1.34)
which applied with (1.27) and (1.31) on the Friedmann equation (1.5), gives us the a-dependent Hubble parameter H(a) = H0 p Ω(a) + (1 − Ω0)a−2 , (1.35) with Ω(a) =X i Ω0ia −3(1+wi) = Ωra−4+ Ωma−3+ Ωxa−3(1+wx) . (1.36)
An equivalent redshift-dependent Hubble parameter can easily be found using (1.14) H(z) = H0
p
Ω(z) + (1 − Ω0)(1 + z)2 . (1.37)
7
Some authors also define a curvature “density” parameter, as Ωk≡ − k H2
0
, and thus Ω0+ Ωk= 1. 8
Lookback time and age of the universe
From the definition of Hubble parameter (1.5) we can derive the expression of the lookback time, i.e. the time interval between the present epoch and the time t of an event that happened at redshift z Z t(a) 0 dt0 = Z a 1 da0 a0H(a0) H0t = Z 1 1+z 0 da h (1 − Ω0) + Ωra−2+ Ωma−1+ Ωxa−(1+3wx) i−12 , (1.38)
where we used (1.35). By setting z → ∞ one obtains the present age of the universe as a function of the densities parameters. Recent measurements (Komatsu et al. 2008) find
t0= 13.73 ± 0.12 Gyr . The (ΩΛ, Ωm) plane 0.0 0.2 0.4 0.6 0.8 1.0 ΩM 0.0 0.5 1.0 1.5 2.0 ΩΛ No Big Bang Accelerating Decelerating Closed Flat Op en
Figure 1.3: The (ΩΛ, Ωm) plane.
In the present day, where the radiation density is very small, we can reduce equation (1.31) to a very simple relation between the energy densities of two components, matter and dark energy. This gives rise to a typical way of plotting our observational constraints, which can be directly compared with predictions from the cosmological models.
A plot of the (ΩΛ, Ωm) space can be seen in Fig.1.3, where we use wΛ= −1. The
CHAPTER 1. BIG BANG COSMOLOGY
1.3.5 Cosmological observables
All the introduced cosmological parameters depend on a certain number of observables, of which we already introduced the redshift. We will now introduce another one: distances. Proper distance
In the comoving frame at which we are at the origin, the distance from us to an object is not an observable. That’s because a distant object can only be observed by the light it emitted at an earlier time, where the universe had a different scale factor, none of which observable either. We need then to define a relationship between the physical distance and the redshift, as a function of the dynamics and the contents of the universe.
The proper distance dPis defined so that 4πd2Pis the area of the sphere over which light from
a far emitting source spreads in the time it travels to us. It can be derived using the metric for a null geodesic Z dP 0 dr √ 1 − kr2 = Z t0 0 dt0 a = Z z 0 dz0 H(z0) = Z 1 a da0 a02H(a0) . (1.39)
Evaluating the left hand side integral and using (1.37) we get dP = 1 H0p|Ωk| X p|Ωk| Z z 0 Z(z0)dz0 , (1.40)
with Ωk= Ω0− 1 (see footnote 7),
X [x] = sin x if k > 0 x if k = 0 sinh x if k < 0 , (1.41) and Z(z0) =hΩr(1 + z0)4+ Ωm(1 + z0)3+ Ωx(1 + z0)3(1+wx)+ Ωk(1 + z0)2 i−12 . (1.42)
If we now have k ∼ 0, (1.40) reduces to (cf. Fig. 1.4) dP= 1 H0 Z z 0 dz0 p Ωr(1 + z0)4+ Ωm(1 + z0)3+ Ωx(1 + z0)3(1+wx) . (1.43) Luminosity distance
The main method to calculate distances to any stellar objects is to estimate its true lumi-nosity and compare that to the observed flux (inversely proportional to the square distance).
The measured flux f is connected to the intrinsic luminosity L by
f = L
4πd2 L
, (1.44)
where dL is the luminosity distance. In an expanding universe, this relation does not hold: the
0.0 0.5 1.0 1.5 2.0 z 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 H0 dP (Ωm, wx) = (0.2,−1.5) (Ωm, wx) = (0.2,−1.0) (Ωm, wx) = (0.3,−1.0) (Ωm, wx) = (0.2,−0.5)
Figure 1.4: Proper distance relation with redshift for different cosmological parameters, in a flat universe.
power of (1 + z) comes from the energy reduction due to wavelength lengthening, and another due to the decreasing frequency on reception, so that
f = L
4πd2
P(1 + z)2
, (1.45)
where dP is the proper distance (1.40). We have thus
dL= dP(1 + z) . (1.46)
In astronomy the brightness of an object is typically expressed in terms of magnitudes. This logarithmic measure of relative flux is so, that the apparent magnitude m of an object is related to its flux f as m − m∗= −2.5 log10 f f∗ = −2.5 log10 L L∗ d2L∗ d2L , (1.47)
and becomes an absolute scale if the object with flux f∗ is used as standard reference. This
reference is an object whose intrinsic luminosity L∗ is such, that if its luminosity distance dL∗
is 10pc, its apparent magnitude m∗ is equal to 0. We can thus define the distance modulus µ
µ ≡ m − M = 5 log10dL 10
= 5 log10[dP(1 + z)] − 5 , (1.48)
with the absolute magnitude
M ≡ −2.5 log10 L L∗
. (1.49)
These definitions imply L∗≈ 78.7L , which is the case for Vega (α Lyr), historically used as
CHAPTER 1. BIG BANG COSMOLOGY
We thus introduced a new observable, the received flux of a distant source. The astronomical object for which we are able to infer its intrinsic luminosity independently of the observed flux is called a standard candle. That’s the case of the Cepheids (used by Hubble for the distances determination), which are variable stars who present a correlation between their period and luminosity, as well of other standardizable objects like globular or galaxies clusters (cf. Roos (2003)).
One important standard candle, observable up to distances where the cosmological redshift effects dominates (and thus for which we can use (1.48) to study the composition of the universe), are the type Ia supernovæ (SNe Ia). They are the main study object of this thesis, and thus the luminosity distance will be our most important observable. The SNe Ia will be further presented in § 2.3.
Angular-diameter distance
The usage of objects of known intrinsic size, the standard rulers, allows us to measure the proper distance in a similar fashion of the luminosity distances, although now in a “orthogonal” direction of the metric. The diameter of a source of light at comoving distance dP is defined by
(1.2) as
D ≡ a(t)θdP . (1.50)
We can then define the angular-diameter distance as dA≡
D
θ = a(t)dP = dP
1 + z , (1.51)
where θ is the measured angular diameter. It relates to the luminosity distance (1.46) as dA=
dL
(1 + z)2 .
A key application of the angular-diameter distance is in the study of features in the cosmic microwave background radiation (§ 2.1).
Other observables
Other observables which I will not detail, include the number of objects (galaxies, quasars) within a comoving volume element, which relates d2P and H(z); and the mass power spectrum from the distribution of galaxies at very large scales, which depends on the initial spectrum of inhomogeneities and their evolution with time. They are mainly used by redshift surveys of large scale structure (§ 2.2).
1.4
The ΛCDM concordance model
We arrive thus at the consensus cosmological model, the so-called ΛCDM (cosmological constant and cold dark matter) model. Considered as the nowadays “standard model” for cosmology, it tells us that:
• the universe is flat (k = 0, Ωo = 1) and had nearly scale-invariant inhomogeneities in
the initial density perturbations (§ 2.1 and 2.2), which are responsible for the structure formation, as predicted by the inflation theory (Guth 2004);
• the total energy density of the universe is accounted for at:
∼ 4% by the baryon energy density, constraint by the big bang model of primordial nucleosynthesis (Burles et al. 2001);
∼ 23% by a cold, nonbaryonic and collisionless matter (cold dark matter);
∼ 73% by a cosmological constant (Λ) type of dark energy (wΛ= −1), which accelerates
the expansion (§2.3).
This model’s predictions and assumptions are currently well constraint by observational data (cf. Chap.2). There’s still however some theoretical animosity towards this model, namely about the cosmological constant. While correctly explaining today’s observations of the accelerated expansion, there is no underlying physics associated to it nor to its constant equation of state. The tentative explanation by means of the vacuum energy (cf. footnote5), struggles with a 120 orders of magnitude difference between the predicted and measured values. Another problem is the so-called coincidence problem: why is Ωm+ ΩΛ' 1 precisely today when we are here to
observe it, after an expansion of several billion years? If Ωm decreases with time and ΩΛremains
constant, why has the cosmological constant begin to dominate the sum only now?
Other than an explanation based on the untestable anthropic principle9, a few modifications to the standard model are proposed to try to solve these and other problems, although they still remain observationally indistinguishable from a cosmological constant model.
1.4.1 Extension models
Most extension models to explain the expansion acceleration can be divided into three cat-egories (Roos 2008) according to their modifications:
dynamical which modify the right hand side of Einstein’s equations (1.3). These quintessence models (Martin 2008) assume a slowly evolving scalar field, with negative pressure, having a potential with an energy density behaving like a decaying cosmological constant. geometrical which modify the left hand side of (1.3). Changes can by made to obtain a new
version of the Friedmann equation which defines a(t), or else to the equations that govern the growth of the density perturbations. These modifications must reduce to the standard GR at very small (solar system), and very large (early times) distances to be compatible with observations.
new spatial conditions by removing the cosmological principle assumption that the universe is spatially homogeneous at large scales. These back-reaction models assume we are at the center of a nearly spherical large scale under-dense region. The non-linear gravitational effects of these spatial density perturbations, averaged over large scales, would give a distance-redshift relation similar to that of an accelerating universe.
9
Observational cosmology
As seen on the previous chapter, the cosmic expansion (and acceleration, which we will see shortly), and the role played by dark energy are one of the most intriguing problems that physics faces today. Recent years have seen a new wealth of experiments arrive, that try to bring answers to these questions, in an unprecedented program to determine the nature of dark energy with high precision.
I will now present the three major axis of experiments that probe the expansion history of the Universe.
2.1
Cosmic Microwave Background
After the Big Bang, the universe was radiation-dominated. In this hot and dense plasma of matter and radiation in thermal equilibrium, photons scattered freely of electrons and prevented the formation of neutral nuclei. With the expansion, the temperature gradually decreased until the photons’ energy was not enough to ionize hydrogen (T ∼ 3000 K), and the neutral light elements appeared. The universe thus became transparent to radiation and the photons were allowed to travel free. This epoch of the last scattering of photons, when the universe had around 380,000 years (z ∼ 1100), is called the photon-decoupling or recombination1 time, and the cosmic microwave background (CMB) is the leftover radiation from this moment.
The CMB was first detected by Penzias and Wilson (1965), and its predicted blackbody spectrum was measured with high accuracy by the COBE (Smoot et al. 1992; Fixsen et al. 1996) space-borne experiment. The measured temperature of the CMB nowadays is
TCMB0 = 2.725 ± 0.002 K .
COBE’s observational breakthrough was the measurement of angular variations in the tem-perature maps of the CMB. These very small anisotropies
∆T T ∼ 10
−5
,
are an expected effect of the inflationary phase, and the key to understand the origin of structure on the universe, since they reflect the departure from homogeneity at the recombination epoch. The anisotropies form from the density perturbations that propagated as acoustic waves in the initial plasma of matter and photons. After decoupling, these perturbations no longer oscillate and the CMB photons have imprinted a snapshot of the state of the fluid at that time.
1
CHAPTER 2. OBSERVATIONAL COSMOLOGY
By analyzing the anisotropies map (Fig. 2.1a), one can extract the fundamental oscillatory mode and corresponding harmonics, and obtain its power spectrum (Fig.2.1b). The exact form of the peaks of the power spectrum depends on the matter content of the universe, thus the measurement of its shape (a standard ruler), yields precise information about many cosmological parameters. These measurements, while not probing dark energy directly, provide constraints like the flatness (Ωk), or the baryon (Ωb) and dark matter densities (Ωc), that can be used to
infer ΩΛ.
(a) (b)
Figure 2.1: (a)WMAP 5 year CMB temperature anisotropies map fromHinshaw et al. (2008);(b)CMB temperature power spectrum from the WMAP 5 year dataset (Nolta et al. 2008), along with results from ACBAR (Reichardt et al. 2008), BOOMERANG (Jones et al. 2006) and CBI (Readhead et al. 2004). The red curve is the best-fit ΛCDM model to the WMAP data.
The most recent CMB measurements come from the WMAP satellite (Komatsu et al. 2008), whose 5 year results for a few ΛCDM parameters, combined with BAO (§2.2) and SNe (§2.3) data, are expressed in Table2.1. In the near future, a new generation of ground and space (like the Planck satellite (The Planck Collaboration 2006)) experiments will further study the CMB with even higher accuracy, searching e.g. for the imprint of primordial gravitational waves in its polarization power spectrum.
Parameter WMAP+BAO+SN t0 13.73 ± 0.12 H0 70.1 ± 1.3 Ωb 0.0462 ± 0.0015 Ωc 0.233 ± 0.013 ΩΛ 0.721 ± 0.015 Ω0 1.0052 ± 0.0064
Table 2.1: Some ΛCDM parameters inferred from the 5 year WMAP+BAO+SN. FromKomatsu et al. (2008).
2.2
Large Scale Structure
result of slight overdensities in the initial baryon-photon plasma, that grew through gravitational instability throughout the expansion.
Figure 2.2: The 2dFGRS map, showing the redshift distribution of the full sample of ∼ 240000 observed galaxies up to z ∼ 0.2. FromColless et al. (2003).
There are several methods which use the LSS to probe for different observables, of which the most promising for dark energy measurements are:
Baryon acoustic oscillations In the same way as the density fluctuations in the primordial photon-baryon plasma show up in the temperature power spectrum of the CMB, they can also be seen in the large scale spatial distribution of baryons. Their manifestation is a spike at ∼ 150 Mpc separation in the matter distribution correlation function, an effective standard ruler and hence a dark energy distance-redshift probe. In the recent years, wide field redshift surveys like the 2dFGRS (Colless et al. 2003) and the SDSS ( Adelman-McCarthy et al. 2008), provided us with 3D maps of the galaxy distribution in the nearby universe. Since galaxies trace the evolution of (dark) matter, these surveys can be used to find the characteristic BAO scale (Eisenstein et al. 2005;Percival et al. 2007).
CHAPTER 2. OBSERVATIONAL COSMOLOGY
2008), to infer the galaxy cluster mass function: direct counting, x-ray flux/temperature, the Sunyaev-Zeldovich effect or weak gravitational lensing.
2.3
Supernovæ
A supernova explosion is the energetic display of a star’s transition to a new phase of its evolution, or its death. This is one of the most energetic events in the universe, creating a new short-lived object in the sky, that outshines its host galaxy during a time-frame of a few weeks. Their extreme intrinsic luminosity make supernovæ observable up to cosmological distances, and thus standardizable probes of dark energy.
I will first present the divisions of the broad supernovæ “label” and corresponding origins, and then detail the present usage of supernovæ in cosmology.
2.3.1 Classification and origins
Historically, supernovæ classification has been based on their spectral features. Minkowski (1941) divided them into two groups, depending on the existence of hydrogen lines on their spectra at maximum light: type I supernovæ do not show H lines, whether type II do. This division was later extended (Filippenko 1997), as it became noticeable that there were more spectral and photometric differences inside each group.
The type I group is divided into 3 sub-families: the type Ia supernovæ (SNe Ia) with strong Si ii absorption lines; and the Ib and Ic, both without Si ii and respectively with or without He I. The type II divides into: II-l and II-p, based on their light curve2 shapes (linear or with a plateau); II-n distinguished by narrow emission lines and slowly decreasing light curves; and II-b which represent a special type of supernova whose early time spectra is similar to type II and late time to Ib/c. This classification and corresponding spectra and light curve examples can be seen in Fig.2.3 and 2.4.
Despite what one could think from this “standard” classification, type Ib/c supernovæ are closer to type II than to type Ia. This can be seen from the similar late time spectra (Fig.2.4a), and from measurements of the explosion rates per host galaxy (Cappellaro et al. 1999): we do not observe any Ib/c or II supernovæ on elliptical galaxies, whether Ia are observed in all galaxy types. Furthermore, SNe Ia present an overall homogeneous spectroscopic and photometric behavior, contrary to the other types. These evidences (as well as the observation of the cross-over II-b type, linking SNe II with SNe Ib/c), reflect the different physical mechanisms, and hence the different origins, between the SNe Ia and all the other SNe.
Assuming a supernova originates from a single stellar object, only two physical mechanisms can explain the observed energy output: either through the release of the nuclear energy by an explosive reaction (thermonuclear supernovæ); either through the release of the gravitational binding energy when a star collapses to a compact object (core-collapse supernovæ).
Thermonuclear supernovæ
The homogeneity of the class of SNe Ia, allied to the facts that no hydrogen or helium exists in their spectra, and no remaining object is found in its remnants, provides a strong hint that their progenitor may be a carbon-oxygen white dwarf (WD) star (Woosley and Weaver 1986).
A WD is the “residual after-life” of stars whose main-sequence mass is lower than 10M .
During the main part of its life, the star burns hydrogen transforming it into helium. The
2
light curve
IIb
IIL
IIP
IIn
ejecta!CSM interactioncore collapse
thermonuclear
yes yes no no noSiII
HeI
yes hypernovae strong shapeIa
Ic
Ib
Ib/c pec
II
I
H
Figure 2.3: Supernovæ classification into different sub-groups. Remark the different origin of the SNe Ia with respect to all the other types. FromTuratto (2003).
maximum 3 weeks one year
!" !# HeI CaII SII SiII [Fe III] [Fe II]+ [Fe III] [Co III] Na I [Ca II] [O I] [O I]
II
Ib
Ic
Ia
!# Na I Ca II !# O I (a) (b)Figure 2.4: (a) Spectra of the main types of supernovæ at three different epochs and characteristic lines; (b) Typical (blue) light curves for the main types of supernovæ. Notice the higher luminosity at maximum of the SNe Ia with respect to all the other types. FromTuratto (2003),Filippenko (1997).
CHAPTER 2. OBSERVATIONAL COSMOLOGY
expand, instead fueling even more the helium burning), increases the temperature continually, and as it approaches the Fermi temperature the electrons become once again non-degenerate and the helium-flash stops.
The star then burns quietly all the helium in the core, creating carbon and oxygen until (as in earlier phases), the core fuel ends and the reactions continue in an outer shell of helium. Once again the star expands, and since the helium burning reaction (triple-alpha) is very sensitive to temperature, the star will enter a pulsating phase, where the balance between internal pressure and gravity changes as the star expands (and the temperature decreases) or contracts (and the temperature increases). These pulsations generate a very strong outflow of mass from the stars surface, which will gradually scatter its envelope in the form of an expanding shell heated by the hot core, a planetary nebula3. The core, constituted of a degenerate carbon and oxygen gas,
continues to contract and cool, and a WD is formed.
The WD’s exotic constitution makes it a strange object: the more massive it is the smaller it gets (a WD with the mass of the Sun would be roughly the size of the Earth). Besides, a WD is only stable up to a specific mass, the Chandrasekhar mass (Chandrasekhar 1931) of ∼ 1.4M ,
above which the degenerate electron gas pressure cannot counter the gravitational collapse. That is the key behind the SNe Ia: if we have a WD in a binary system (the single-degenerate scenario), accreting matter from a companion (Fig.2.5), its mass will steadily increase up to the Chandrasekhar limit. At that moment, the density in the core is such that carbon fusion ignites, leading to the production of 56Ni. As we have already seen, thermonuclear fusion is unstable in the degenerate matter, in the sense that it does not allow the moderation of the burning by expansion. A nuclear instability ensues, followed by a nuclear burning front that propagates through the star and explodes it completely.
A new SNe Ia then appears in the sky and its luminosity peaks in just a few days. The consequent decrease of the light curve over several months cannot be powered by the explosion itself, since temperature decreases too fast, and is explained instead by the radioactive decay of the56Ni produced during the carbon fusion, into its daughter nucleus: 56Ni →56Co →56Fe.
While accounting for the relative SNe Ia homogeneity (related to the limit imposed by the Chandrasekhar mass), not all the physical processes behind this model are yet fully un-derstood. Several flame propagation and explosion scenarios are proposed (Hillebrandt and Niemeyer 2000), and despite the huge improvement in recent years, numerical hydrodynamical simulations of SNe Ia explosions are still incapable of providing an accurate enough description of observations (Roepke 2008).
Core-collapse supernovæ
The fact that type II or Ib/c supernovæ are absent from the elliptical galaxies, which contain only old, low-mass stars, makes a strong point on their origin as the gravitational collapse of a massive star. In the core of their progenitors (stars with mass higher than ∼ 10 M ), the
hydrostatical equilibrium holds as long as the fusion reactions burn the available hydrogen and form heavier and heavier elements, up to iron. The star has then an “onion-layer” structure, with layers of different composition, from the iron in the core to the hydrogen4 in the envelope. Since no further exothermic reactions are possible, as the iron core mass continues to increase, it reaches the Chandrasekhar mass and the internal (degenerate electron) pressure can no longer counterbalance gravity. The star collapses until the core attains a large enough density and
3
Historically called this way, since it looks like a planet when viewed through a small telescope. 4
Figure 2.5: Artist impression of a SNe Ia caused by the explosion of a WD accreting matter in a binary system. By D. Hardy (http://www.astroart.org).
temperature, where the iron nucleons are photo-dissociated into helium nuclei and free electrons, which (through inverse β decay) combine with protons to form neutrons and neutrinos. A neutron star5 is formed. The sudden increase in pressure, due to the strong-force bounded neutrons, make the collapsing layers rebound, and the radiation pressure of the huge neutrino flow from the core would then be responsible for the displacement of the outer remaining layers of the star, which is then observed as a supernova.
The observation of supernova remnants with neutron stars in their center further supports this model.
2.3.2 SNe Ia homogeneity - standard candles
Visible up to great distances (the furthest observed until today has z = 1.75), supernovæ are a good candidate for a cosmological probe. Contrary to SN II and Ib/c, whose energy output is heavily dependent on the (wide-ranging) mass of the progenitor, SNe Ia have a similar evolution and explosion process (the thermonuclear disruption of a WD near the Chandrasekhar mass), and we can thus expect them to show a small scatter around their intrinsic peak luminosity. If this assumption is correct, SNe Ia explosions provides us with a good standard candle for distances way beyond the Cepheids. So, how good is this homogeneity?
Photometrically
The most common (and less telescope-time consuming) way to observe supernovæ is by photometric followup of their evolution. This is done by measuring on consecutive nights its luminosity (apparent magnitude), thus obtaining light curves in different filters6. The light curves for a typical SN Ia are shown in Fig.2.6a.
5
Under certain conditions, this neutron star can further collapse into a black hole. 6
CHAPTER 2. OBSERVATIONAL COSMOLOGY
0 50 100
Days from Bmax
18 16 14 12 10 8 Magnitude + const U+1 B V-1 R-2 I-3 J-5 H-6 K-7 (a) −20 −19 −18 −17 −20 −19 −18 −17
M
max−
5log(H
o/65)
−20 −19 −18 −17 1.0 1.5 2.0B
V
II
Δm
15(B)
obs (b)Figure 2.6: (a)Light curve in several bands (Johnson-Cousins UBVRI and near-infrared) from a typical SN Ia (SN2003du). Phase is with relation to B maximum. Adapted from Stanishev et al. (2007); (b)
Brigther-slower correlation for SNe Ia light curves. Adapted fromPhillips et al. (1999).
On the other hand, if one wants to know their absolute magnitude (to test our standard candle assumption), the case is somewhat different, since we need to have a measure of its distance. Hamuy et al. (1996) observed a number of nearby supernovæ in the smooth Hubble flow7, for which the distance can be estimated via the host galaxy redshift and Hubble’s law. The found (uncorrected - see next paragraph) values present some dispersion (Fig.2.7a): scatters of 0.38 and 0.26 magnitudes for MB and MV (the absolute magnitude in the B and V bands
respectively) were found. These scatters are compatible with results from several teams, which have been using observations of SNe Ia on galaxies with distances known through different techniques, like Cepheids measurements, for the determination of the Hubble constant: Sandage et al. (2006)observes mean absolute magnitudes MBV between -19.3 and -19.55, compatible with
the 0.3 magnitudes range span of the MB measured by other teams, depending on calibration
techniques and observational methods (see Table 2 inGibson et al. (2000)).
Despite the existence of this scatter (thought to be mainly due to the amount of 56Ni pro-duced during the explosion (Timmes et al. 2003)), it is remarkably small: SNe Ia seem indeed to be good standard candles.
The first step in the SNe Ia standardization was made when, as new supernovæ started
7
to be systematically observed, it became clear that a linear correlation exists between the peak luminosity and the decline rate of their light curve. This “brighter-slower” (i.e. brighter objects’ light curves seem to decline slower, see Fig.2.7a) relation was introduced byPhillips (1993), in the form of the parameter ∆m15, which is the difference in magnitudes between the maximum
and 15 days after it (Fig. 2.6b). By applying this empirical correction model to the absolute magnitudes, Hamuy et al. (1996) found the scatter reduced by a factor of 2, which entails a distance determination precision of 7 − 10%.
Several other standardization methods followed, notably: the MLCS (Riess et al. 1996) and stretch (Perlmutter et al. 1997b;Goldhaber et al. 2001) (Fig. 2.7b) methods, using supernovæ light curve shape templates; the ∆C12 (Wang et al. 2005) method for the “brighter-bluer”8
correlation; the MLCS2k2 (Jha et al. 2007) and SALT[2] (Guy et al. 2005; Guy et al. 2007) methods, using both light curve shape and color parameters. In this work we will be mostly interested in the SALT approach, further detailed in § 2.3.4.
(a) (b)
Figure 2.7: C´alan/Tololo nearby supernovæ absolute magnitudes in V band, (a) as observed and(b)
after correction using the stretch parameter. Adapted fromPerlmutter et al. (1997a).
It should be noticed that all these methods are mainly empirical. They correlate a distance-dependent parameter with one or more distance-indistance-dependent parameters, and are applied on the single evidence that they improve the fit around the Hubble line in the Hubble diagram (cf. § 2.3.4). This approach is valid for a cosmological analysis, where we are interested in the relative distances between nearby and distant supernovæ sets, both standardizable by the most recent methods.
Spectroscopically
“Branch-normal” (Branch et al. 2006) SNe Ia optical spectra (Fig.2.8a) present a remarkable homogeneity, as can be seen in Fig. 2.8b. The most identifiable spectral characteristic of a SN Ia (and its defining feature) is the prominent absorption around 6100˚A, due to the Si ii doublet (λλ6347˚A and 6371˚A). Other observable lines near maximum light are Ca ii (λλ3934˚A, 3968˚A, and λ8579˚A), Si ii (λ3858˚A, λ4130˚A, λ5051˚A, λ5972˚A), Mg ii (λ4481˚A), S ii (λ5468˚A and λλ5612˚A, 5654˚A) and O i (λ7773˚A). The spectrum is also scattered with low-ionization Ni, Fe, and Co lines which increase in later times.
8
CHAPTER 2. OBSERVATIONAL COSMOLOGY 4000 5000 6000 7000 Rest Wavelength (Å) 0 1 2 3 4 Scaled f λ + Constant SN 2001V day +20 Si II Si II S II Fe II Si II Fe III Fe II Si II Ca II Ca II Fe II Co II Cr II Fe IICr IICo II Cr II Na I Si IIFe II (a) 4000 5000 6000 7000 8000 WAVELENGTH 1 10 100 FLUX 96X 94ae 90N 98bu 01el 94D 98aq (b)
Figure 2.8: (a) Spectrum and characteristic lines at maximum (up) and 20 days later (bottom) for a typical SN Ia (SN1998aq). FromMatheson et al. (2008). (b)Spectra at maximum luminosity for several “Branch-normal” SNe Ia. FromBranch et al. (2006).
The analysis of SNe Ia spectral evolution divides their lifetime into two phases: the early photospheric phase, where the outer layers are opaque to radiation and we observe their consti-tution in the form of absorption lines, that change as the layers expand and become transparent; and the nebular phase, several weeks after maximum, when the environment has become trans-parent and we observe the emission lines of the inner layers, that can be attributed to Co and Fe transitions. The nebular phase is dominated by the changing strength of these individual lines multiplets. Despite the overall uniformity, subtle differences exist in lines strengths and velocities between different SNe Ia (Hachinger et al. 2006).
Deserving to be placed in a category apart, are a lot of peculiar SNe Ia presenting very different light curves and spectra from the “typical” ones. These will be presented in § 2.3.3.
While this photometric/spectroscopic division was made for easier presentation, they are obviously intrinsically connected: a light curve point is nothing more than the integration of a supernova spectrum by a bandpass filter. It is then clear that we could easily study the homogeneity and standardization of the SNe Ia class by simply using absolutely flux calibrated spectra. We will see in § 3how the SNfactory project intends to do this.
2.3.3 SNe Ia heterogeneity - peculiar objects
